Project Description:

MODELING FLOW IN VUGGY MEDIA

Todd Arbogast, Mathematics

Steve Bryant, Petroleum & Geosystems Engineering

Jim Jennings, Bureau of Economic Geology

Charlie Kerans, Bureau of Economic Geology

The University of Texas at Austin

Summer School in

Geophysical Porous Media:

Multidiscplinary Science from Nano-to-Global-Scale

July 17-28, 2006

Purdue University, West Lafayette, Indiana

Center for Subsurface ModelingInstitute for Computational Engineering and Sciences

The University of Texas at Austin, USA

Vuggy Porous Media

Pipe Creek Reef Outcrop

Pipe Creek Reef outcrop, Red Bluff River near Pipe Creek, Texas.

River basin area of the Glen Rose formation, looking upstream.

Pipe Creek Reef Outcrop

Wide view of region with both in-situ fossils and fossil debris.

Rudist Caprinids at Pipe Creek Reef

Illustration of a typical 4-6 cm diameter

Cretaceous (Albian) caprinid fossil.

Pipe Creek Reef Outcrop

Closeup of in-situ caprinids (found as in original colonies)

Pipe Creek Reef Outcrop

Closeup of caprinid fossils within a debris region.

Pipe Creek Reef Outcrop

Closeup of caprinid fossils within a debris region,

showing the internal structure of the shells.

Pipe Creek Reef Mounds

Conceptual stochastic model of mound cores (left) and

debris deposits (right) in a one square kilometer segment

of a patch reef belt in the Pipe Creek area based upon

outcrop observations and interpretations from 18 cored

wells (factor of 10 vertical exaggeration).

Experimental Wells

Two wells were drilled approximately 5 feet apart

to a depth of about 20 feet.

Core Sample

A core sample from the formation.

Main Questions

How does fluid flow in such a system?

1. Are the vugs interconnected? If so, over what distances?

• The well test suggests that they are connected for at least 5 feet.

• Fluid flows much more rapidly in an interconnected vug system

than in ordinary porous medium.

2. Is there a well defined structure to the vug network?

3. Is there an effective permeability for the system?

• It appears so, but finding it is problematic.

4. How do tracers (i.e., chemicals) transport in such a system?

• They exhibit exotic behavior.

Geometric Analysis of a Large Sample

Pipe Creek Sample Location

Closeup of debris region from where first large sample was taken.

Internal Structure of the Vuggy Network

CT Scans of Pipe Creek Reef sample (about 30 × 30 × 36 cm3)

240 slices 1.5 mm thick with 512 × 512 pixels 0.547 × 0.547mm square

Interior Vug Network

Reconstructed vug network from CT scans.

Tracer Experiments

Experimental Design

Effluent

Rock

Water

or

Tracer

Time

TracerConcentration

Injection history (slug injection).

Typical Experimental Results

0.0

0.2

0.4

0.6

0.8

1.0

Effluentconcentration

0 4 8 12 16 20

Pore volumes injected

Early breakthrough

Multiple plateaus

Abrupt drop

Single plateau

Long tail

Typical effluent concentration history.

Note that the curve has five definite features.

Theoretical Micro-Model

The Governing Equations

Notation:

• D is the symmetric gradient: (Du)i,j =1

2

(

∂ui

∂xj+

∂uj

∂xi

)

• µ is the fluid viscosity

• K is the rock permeability

• f is a gravity term

• q is a source term

Vuggy region: Ωs

−2µ∇ · Du + ∇p = f Stokes Equations

∇ · u = q Conservation

Rock matrix: Ωd

µK−1u + ∇p = f Darcy Law

∇ · u = q Conservation

The Interface and Boundary Conditions

Notation:

• ν is the outer unit normal vector

• τ is a unit tangent vector

• α is the Beavers and Joseph slip coefficient

Darcy-Stokes Interface: Γ = ∂Ωs ∩ ∂Ωd

us · νs = ud · νs Continuity of flux

2νs · Dus · τ = −α√K

us · τ Beavers-Joseph-Saffman slip condition

2µνs · Dus · νs = ps − pd Continuity of normal stress

External Boundary: ∂Ω

us = 0 on ∂Ω ∩ ∂Ωs No slip

ud · ν = 0 on ∂Ω ∩ ∂Ωd No normal flow

Simplification to Pipe Flow

The flow in the matrix rock is very slow, so ignore it. We are left with

Stokes flow in a channel, which is like a pipe.

We can solve the Stokes equations for pipe flow,

−2µ∇ · ∇u + ∇p = 0,

∇ · u = 0,

when the domain is simple, by exploiting symmetries.

`00

d

u = 0

u = 0

p = P p = 0

This is a rectangle in 2-D but a cylinder in 3-D.

It turns out that the average velocity is proportional to the pressure

gradient P/`, which gives us the permeability as a function of the

diameter d, for a pipe-like vug.

Caution: But this is not what we have in the rock!

Potential Project Activities

Data Available

• A data file of CT scan pixels converted into a index: vug, rock, or

“surface mud.”

• Overall porosity ≈ 20%.

• Permeability data

• Matrix rock permeability ≈ 10 mD

• Lab test of the large sample effective permeability ≈ 100 Darcy

• Lab test of the well core effective permeability ≈ 1 Darcy.

• Field well test effective permeability ≈ 1 Darcy over 5 feet.

• Tracer slug test typical behavior.

Questions and Activities—1

Progress can probably be made on the following within two weeks.

1. Determine the vug structure of the large sample, and show that it is

interconnected. This could be approached by analyzing the pixel

data. It would involve writing a data analysis code.

• Two vug pixels that share a face are most likely interconnected.

• What about vug pixels that share only an edge or a point?

• What about dead end channels?

2. Can we treat the system as a Darcy system with a large permeability

in the vugs? This could involve solving single phase flow problems

for different “large” vug permeabilities.

• Should we take the vug permeability to be infinity?

• Is the system sensitive to the assumed vug permeability?

• Can we determine the effective permeability of a sample this way?

• Do ideas from pipe flow help here?

3. How does the vug network structure affect effective permeability?

This could involve treating the system as a Darcy system with

different patterns of vug channels.

• Corners? Branchings? Constrictions? Pinchouts? Dead ends?

Questions and Activities—2

The following may be too ambitious for the two week project period.

4. Can we explain the features in the tracer experimental data?

• Postulate a reason for the various features.

• Run simulations to test the hypotheses. This involves running flow

and transport codes.

Contacts

Todd Arbogast.

I am here through Monday, July 24 (I leave Tuesday morning).

Steve Bryant.

Available to answer questions by e-mail for the full two weeks:

Steven Bryant@mail.utexas.edu